## The Combinatorics of the leading root of Ramanujan’s function

#### When/Where:

August 21, 2015, 1:55 — 2:45pm at LIT 368.

#### Abstract:

I consider the leading root \(x_0(q)\) of Ramanujan’s function

\(\sum\limits_{n=0}^\infty\frac{(-x)^nq^{n^2}}{(1-q)(1-q^2)\ldots(1-q^n)}\).

I prove that its formal power series expansion

\(qx_0(-q)=1+q+q^2+2q^3+4q^4+8q^5+\ldots\)

has positive integer-valued coefficients, by giving an explicit combinatorial interpretation

of these numbers in terms of trees whose vertices are decorated with polyominos.

Similar results are obtained for the leading roots of the partial Theta function and the

Painleve Airy function.